y = px + f(p),
where p is an abbreviation for dy/dx; this is known as Clairaut's
form. By differentiation in regard to x it gives
dp dp
p = p + x-- + f'(p)--,
dx dx
where
d
f'(p) = -- f(p);
dp
thus, either (i.) dp/dx = 0, that is, p is constant on the curve
satisfying the differential equation, which curve is thus any one of
the straight lines y = cx = f(c), where c is an arbitrary constant, or
else, (ii.) x + [f]'(p) = 0; if this latter hypothesis be taken, and p
be eliminated between x + f'(p) = 0 and y = px + f(p), a relation
connecting x and y, not containing an arbitrary constant, will be
found, which obviously represents the envelope of the straight lines y
= cx + f(c).
In general if a differential equation [phi](x, y, dy/dx) = 0 be
satisfied by any one of the curves F(x, y, c) = 0, where c is an
arbitrary constant, it is clear that the envelope of these curves,
when existent, must also satisfy the differential equation; for this
equation prescribes a relation connecting only the co-ordinates x, y
and the differential coefficient dy/dx, and these three quantities are
the same at any point of the envelope for the envelope and for the
particular curve of the family which there touches the envelope. The
relation expressing the equation of the envelope is called a
_singular_ solution of the differential equation, meaning an
_isolated_ solution, as not being one of a family of curves depending
upon an arbitrary parameter.